Las bases de los pronósticos
Pablo Benavides-Herrera
2020-04-06
Las paqueterías a usar en este documento:
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All packages loaded successfullyUn flujo de trabajo limpio de pronóstico
El flujo de trabajo, cuando se va a realizar un proceso, se puede dividir en pasos.
- Preparación de los datos (limpieza)
- Gráfica de los datos (visualización)
- Definición del modelo (especificación)
- Entrenamiento del modelo (estimación)
- Revisar el desempeño del modelo (evaluación)
- Producir pronósticos
1. Preparación de los datos (limpieza)
Esto siempre es lo primero que se hace y puede ir desde simplemente cargar los datos en R, hasta limpiezas más detalladas, como identificar valores omitidos, NA, filtrado de la serie, etc. Para esto, utilizamos varias funciones de las paqueterías tsibble y tidyverse, que simplifican mucho estas tareas.
2. Gráfica de los datos (visualización)
Continuaremos trabajando con los datos de global_economy para ejemplificar esto.
global_economy %>%
filter(Country == "Sweden") %>%
autoplot(GDP) +
ggtitle("PIB de Suecia") + ylab("$US billions")
3. Definición del modelo (especificación)
Antes de ajustar un modelo a los datos, se debe describir el modelo. Existen muchos tipos de modelos de pronóstico distintos, y es muy importante que escojamos el más apropiado, para obtener buenos pronósticos.
Los modelos en R, como vimos con la regresión lineal (lm(y ~ x1 + x2 + ... + xn)), se especifican en formato de fórmula ( y ~ x), siendo la y la variable dependiente (o variable a explicar), y la o las x las variables independientes (o variables explicativas, regresoras, etc.).
Podemos tomar, p. ej., un modelo lineal de series de tiempo, TSLM, modela los datos que se le incluyan mediante una tendencia lineal.
TSLM(GDP ~ trend())4. Entrenamiento del modelo (estimación)
Una vez que se especificó el modelo, lo que sigue es entrenar al modelo.
Entrenar un modelo significa pasarle los datos para que, estadísticamente, encuentre los parámetros que realizan el mejor ajuste posible.
Siguiendo con el ejemplo del modelo lineal:
fit <- global_economy %>%
model(Modelo_tendencia = TSLM(GDP ~ trend()))Con esto se ajustó un modelo lineal y el objeto resultante es un mable (model table).
fit# A mable: 263 x 2
# Key: Country [263]
Country Modelo_tendencia
<fct> <model>
1 Afghanistan <TSLM>
2 Albania <TSLM>
3 Algeria <TSLM>
4 American Samoa <TSLM>
5 Andorra <TSLM>
6 Angola <TSLM>
7 Antigua and Barbuda <TSLM>
8 Arab World <TSLM>
9 Argentina <TSLM>
10 Armenia <TSLM>
# ... with 253 more rows5. Revisar el desempeño del modelo (evaluación)
Ya teniendo el modelo entrenado, debemos revisar el performance en los datos reales. Esto es, ¿qué tan bien se ajusta el modelo a los datos?, en caso de estar indecisos entre varios modelos, ¿cuál escogemos como el mejor y por qué?
6. Producir pronósticos
Cuando ya evaluamos que el modelo ajustado se encuentra dentro de los parámetros deseados, podemos proceder a realizar los pronósticos. En .R, podemos usar el comando forecast(), en el cual debemos especificar el número de periodos a pronosticar. Por ejemplo, par pronosticar los siguientes 12 meses, escribiríamos h = 12. También podemos usar lenguaje ordinario (en inglés), h = "1 year".
fcst <- fit %>% forecast(h = "3 years")
fcst# A fable: 789 x 5 [1Y]
# Key: Country, .model [263]
Country .model Year GDP .mean
<fct> <chr> <dbl> <dist> <dbl>
1 Afghani~ Modelo~ 2018 N(1.6e+10, 1.3e+19) 1.62e10
2 Afghani~ Modelo~ 2019 N(1.7e+10, 1.3e+19) 1.65e10
3 Afghani~ Modelo~ 2020 N(1.7e+10, 1.3e+19) 1.68e10
4 Albania Modelo~ 2018 N(1.4e+10, 3.9e+18) 1.37e10
5 Albania Modelo~ 2019 N(1.4e+10, 3.9e+18) 1.42e10
6 Albania Modelo~ 2020 N(1.5e+10, 3.9e+18) 1.46e10
7 Algeria Modelo~ 2018 N(1.6e+11, 9.4e+20) 1.58e11
8 Algeria Modelo~ 2019 N(1.6e+11, 9.4e+20) 1.61e11
9 Algeria Modelo~ 2020 N(1.6e+11, 9.4e+20) 1.64e11
10 America~ Modelo~ 2018 N(6.8e+08, 1.7e+15) 6.82e 8
# ... with 779 more rowsEl resultado es una fable (forecasting table) o tabla de pronósticos. El pronóstico se puede graficar fácilmente junto con los datos reales, usando autoplot().
fcst %>%
filter(Country=="Sweden") %>%
autoplot(global_economy) +
ggtitle("PIB de Suecia") + ylab("$US billions")
Métodos sencillos de pronóstico
Utilizaremos de benchmark a lo largo del curso estos métodos básicos de pronóstico. En ocasiones, a pesar de su sencillez, pueden llegar a ser muy útiles.
Utilizaremos los datos de producción de ladrillos para esta sección.
bricks <- aus_production %>% filter_index("1970" ~ "2004")
bricksbricks %>% autoplot(Bricks)
Método del promedio (media)
En este pronóstico, las predicciones de todos los valores futuros son la media de los datos históricos.
\[ \hat{y}_{T+h | T}=\bar{y}=\left(y_{1}+\cdots+y_{T}\right) / T \]
bricks %>% model(MEAN(Bricks))# A mable: 1 x 1
`MEAN(Bricks)`
<model>
1 <MEAN>Método ingenuo (Naïve method)
Aquí, lo que se hace es que se toma el último valor como el pronóstico para todos los valores futuros.
\[ \hat{y}_{T+h | T}=y_{T} \] Dado que un pronóstico ingenuo es óptimo cuando se tienen datos que siguen una caminata aleatoria, a estos pronósticos se les conoce como pronósticos de caminata aleatoria.
bricks %>% model(NAIVE(Bricks),
RW(Bricks)) # hace exactamente lo mismo que NAIVE()# A mable: 1 x 2
`NAIVE(Bricks)` `RW(Bricks)`
<model> <model>
1 <NAIVE> <NAIVE>tidyquant::tq_get("AAPL") %>%
timetk::plot_time_series(.date_var = date,.value = close, .smooth = FALSE)Warning: `type_convert()` only converts columns of type 'character'.
- `df` has no columns of type 'character'Método ingenuo estacional (seasonal Naïve)
Un método similar es el ingenuo estacional. Lo que cambia con el anterior es que se agrega un componente para lidiar con datos altamente estacionales.
\[ \hat{y}_{T+h | T}=y_{T+h-m(k+1)} \]
bricks %>% model(SNAIVE(Bricks))# A mable: 1 x 1
`SNAIVE(Bricks)`
<model>
1 <SNAIVE>vic_elec %>%
autoplot(Demand) %>%
plotly::ggplotly()vic_elec %>%
model(SNAIVE(Demand)) %>%
forecast(h = "1 day") %>%
autoplot(vic_elec %>% filter_index("2014-12-30" ~ .))
Método del drift (deriva)
Este método es una variación del método ingenuo, que permite que el pronóstico aumente o disminuya en el tiempo. El aumento del cambio es el cambio promedio en los datos históricos.
\[ \hat{y}_{T+h | T}=y_{T}+\frac{h}{T-1} \sum_{t=2}^{T}\left(y_{t}-y_{t-1}\right)=y_{T}+h\left(\frac{y_{T}-y_{1}}{T-1}\right) \]
bricks %>% model(
RW(Bricks ~ drift())
)# A mable: 1 x 1
`RW(Bricks ~ drift())`
<model>
1 <RW w/ drift>Esto es lo mismo que trazar una línea recta que conecte el primer y último punto en los datos históricos y continuar la recta hacia adelante.
# Set training data from 1992 to 2006
train <- aus_production %>% filter_index("1992 Q1" ~ "2006 Q4")
# Fit the models
beer_fit <- train %>%
model(
Mean = MEAN(Beer),
`Naïve` = NAIVE(Beer),
`Seasonal naïve` = SNAIVE(Beer),
Drift = RW(Beer ~ drift())
)
# Generate forecasts for 14 quarters
beer_fc <- beer_fit %>% forecast(h= 14)
# Plot forecasts against actual values
beer_fc %>%
autoplot(filter_index(aus_production, "1992 Q1" ~ .), level = NULL) +
ggtitle("Forecasts for quarterly beer production") +
xlab("Year") + ylab("Megalitres") +
guides(colour=guide_legend(title="Forecast")) +
geom_vline(xintercept = as.Date("2007-01-01"), color = "firebrick",
linetype = "dashed") +
annotate("label", x = c(as.Date("2003-01-01"),as.Date("2009-01-01")),
y = 550, label = c("Train set", "Test set"),
color = c("black","blue"))
beer_fc %>%
filter(.model == "Seasonal naïve") %>%
autoplot(filter_index(aus_production, "1992 Q1" ~ .)) +
ggtitle("Seasonal naïve forecast for quarterly beer production") +
xlab("Year") + ylab("Megalitres") +
guides(colour=guide_legend(title="Forecast")) +
geom_vline(xintercept = as.Date("2007-01-01"), color = "firebrick",
linetype = "dashed") +
annotate("label", x = c(as.Date("2003-01-01"),as.Date("2009-01-01")),
y = 550, label = c("Train set", "Test set"),
color = c("black","blue"))
Otro ejemplo:
gafa_stock %>% distinct(Symbol)# Re-index based on trading days
google_stock <- gafa_stock %>%
filter(Symbol == "GOOG") %>%
mutate(day = row_number()) %>%
update_tsibble(index = day, regular = TRUE)
# Filter the year of interest
google_2015 <- google_stock %>% filter(year(Date) == 2015)
# Fit the models
google_fit <- google_2015 %>%
model(
Mean = MEAN(Close),
`Naïve` = NAIVE(Close),
Drift = NAIVE(Close ~ drift()),
SNAIVE = SNAIVE(Close)
)Warning: 1 error encountered for SNAIVE
[1] Non-seasonal model specification provided, use RW() or provide a different lag specification.google_fit# Produce forecasts for the 19 trading days in January 2015
google_fc <- google_fit %>% forecast(h = 19)
google_fc# A better way using a tsibble to determine the forecast horizons
google_jan_2016 <- google_stock %>%
filter(yearmonth(Date) == yearmonth("2016 Jan"))
google_fc <- google_fit %>% forecast(google_jan_2016)
# Plot the forecasts
google_fc %>%
autoplot(google_2015, level = NULL) +
autolayer(google_jan_2016, Close, color='black') +
ggtitle("Google stock (daily ending 31 Dec 2015)") +
xlab("Day") + ylab("Closing Price (US$)") +
guides(colour=guide_legend(title="Forecast"))Warning: Removed 19 row(s) containing missing values (geom_path).
p1 <- google_fc %>%
filter(.model == "Drift") %>%
autoplot(google_2015) +
autolayer(google_jan_2016, Close, color='black') +
ggtitle("Google stock (daily ending 31 Dec 2015)") +
xlab("Day") + ylab("Closing Price (US$)") +
guides(colour=guide_legend(title="Forecast"))
p2 <- google_fc %>%
filter(.model == "Naïve") %>%
autoplot(google_2015) +
autolayer(google_jan_2016, Close, color='black') +
ggtitle("Google stock (daily ending 31 Dec 2015)") +
xlab("Day") + ylab("Closing Price (US$)") +
guides(colour=guide_legend(title="Forecast"))
p1 / p2
Valores ajustados (fitted) y residuales
Cada observación en una serie de tiempo puede ser pronosticada utilizando los datos históricos previos. A estos se les conoce como valores ajustados (o fitted), \(\hat{y}_t\).
Los residuales en un modelo de series de tiempo es la información que el modelo no logró capturar. Esto es, es la diferencia entre los valores reales y los valores ajustados.
\[ e_{t}=y_{t}-\hat{y}_{t} \]
En R, podemos obtener los valores ajustados y los residuales con la función augment(). Recordando, habíamos ajustado tres modelos distintos, que guardamos en la variable beer_fit.
?augmentaugment(beer_fit)Es muy importante analizar los residuos para determinar si nuestros modelos están bien ajustados. Si logramos detectar patrones en los residuales, puede ser indicio de que el modelo puede mejorarse.
Diagnóstico de residuales
Un buen modelo de pronóstico va a producir residuales con las siguientes características:
No están autocorrelacionados. Si se detectan correlaciones entre residuos, todavía hay información útil que se debe modelar.
La media de los residuos es cero. Si la media es distinta de cero, entonces el pronóstico está sesgado.
Nota: El hecho de que un pronóstico cumpla esto, no quiere decir que sea el mejor pronóstico que podamos hacer. Próximamente revisaremos qué otras medidas podemos evaluar para determinar cuál es el mejor pronóstico.
Existen dos características adicionales que son útiles, mas no necesarias, para los residuos de un pronóstico:
Los residuos tienen una varianza constante.
Los residuos se distribuyen de manera normal.
Las transformaciones de Box-Cox pueden ayudar, en algunos casos a lograr cumplir estas características.
Continuemos con el ejemplo del pronóstico del precio de la acción de Google. En muchas ocasiones, el mejor pronóstico para los precios de mercados bursátiles e índices suele ser el realiado mediante el método Naïve.
google_2015 %>% autoplot(Close) +
xlab("Day") + ylab("Closing Price (US$)") +
ggtitle("Google Stock in 2015")
Aplicaremos solo el método Naïve para pronosticar el precio futuro de la acción de Google.
aug <- google_2015 %>%
model(NAIVE(Close)) %>%
augment()
aug
# aug %>%
# features(.resid, mean(.,na.rm = TRUE))
aug %>% pull(.resid) %>% mean(na.rm = TRUE) [1] 0.9439931aug %>% autoplot(.resid) + xlab("Día") + ylab("") +
ggtitle("Residuales del método naïve")Warning: Removed 1 row(s) containing missing values (geom_path).
De la gráfica de los residuales podemos observar que la media parece estar muy cercana al cero y que la variación parece invariante en el tiempo, a excepción de un outlier.
aug %>%
ggplot(aes(x = .resid)) +
geom_histogram() +
ggtitle("Histograma de los residuales")`stat_bin()` using `bins = 30`. Pick better value
with `binwidth`.
Del histograma, podemos ver que los residuales parecen distribuirse como una normal, pero con una cola más grande.
aug %>% ACF(.resid)
aug %>% ACF(.resid) %>% autoplot() + ggtitle("ACF of residuals")
google_2015 %>%
model(NAIVE(Close)) %>%
gg_tsresiduals()Warning: Removed 1 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values
(geom_point).
Warning: Removed 1 rows containing non-finite values
(stat_bin).
aus_production %>%
filter_index("1992" ~ .) %>%
model(SNAIVE(Beer)) %>%
gg_tsresiduals()Warning: Removed 4 row(s) containing missing values (geom_path).
Warning: Removed 4 rows containing missing values
(geom_point).
Warning: Removed 4 rows containing non-finite values
(stat_bin).
aus_production %>%
filter_index("1992" ~ .) %>%
gg_tsdisplay(Beer)
La función de autocorrelación muestra que los residuos no están autocorrelacionados.
Así, el método naïve parece generar pronósticos que capturan toda la información relevante de la serie y, por lo tanto, que satisfacen todas las características necesarias. Por consecuencia, los pronósticos derivados de este método pueden ser bastante buenos, pero los intervalos de predicción pudieran ser imprecisos.
Podemos obtener estas mismas gráficas con un solo comando, gg_tsresiduals().
google_2015 %>%
model(NAIVE(Close)) %>%
gg_tsresiduals() +
ggtitle("Diagnóstico de residuales para el modelo Naïve")
Como vimos antes, el método Naïve es óptimo para este tipo de series. ¿Qué hubiera sucedido si ajustáramos otro modelo? Probemos con el método de la media:
google_2015 %>%
model(MEAN(Close)) %>%
augment() %>%
pull(.resid) %>%
mean(na.rm = TRUE)[1] 2.710222e-14google_2015 %>%
model(MEAN(Close)) %>%
gg_tsresiduals() +
ggtitle("Diagnóstico de residuales para el modelo de Media")
Observamos que estas gráficas tienen un comportamiento muy distinto al Naïve:
En la gráfica de los residuales vemos que se distingue claramente un patrón. De hecho, es el mismo patrón exactamente que siguen los datos originales, restándoles su media.
La función de autocorrelación tiene un comportamiento típico de una caminata aleatoria. Por lo tanto, las autocorrelaciones son significativas.
El histograma de los residuos muestra claramente que no se distribuyen de manera normal.
Tests de Portmanteau de autocorrelación
Para analizar de manera más formal la presencia o ausencia de autocorrelación en los residuos, podemos realizar estas pruebas estadísticas para determinar si las primeras \(h\) autocorrelaciones son significativamente distintas de cero o no.
Test de Box-Pierce
\[ Q=T \sum_{k=1}^{h} r_{k}^{2} \] En este test, \(h\) es el rezago máximo a considerar y \(T\) es la cantidad de observaciones en la muestra.
Si cada \(r_{k}^{2}\) es pequeña, entonces \(Q\) será pequeña. Se sugiere utilizar \(h = 10\) para datos no estacionales y \(h = 2m\) para datos estacionales (donde \(m\) es el periodo estacional). Sin embargo, la prueba no es tan buena cuando \(h\) es grande, relativamente (o sea, cuando \(h\) es mayor a \(T/5\)). En esos casos, es mejor utilizar \(h = T/5\).
Test de Ljung-Box
Un test relacionado y que, generalmente, es más preciso es el test de Ljung-Box.
\[ Q^{*}=T(T+2) \sum_{k=1}^{h}(T-k)^{-1} r_{k}^{2} \] En este caso es igual: valores grandes de \(Q^{*}\) son indicios de que las autocorrelaciones no provienen de ruido blanco.
Entonces, la hipótesis nula de estas pruebas es que la serie en cuestión no está autocorrelacionada. En otras palabras, la \(H_0\) dice que la serie es ruido blanco. Si \(\alpha\) es el nivel de significancia (el nivel máximo de error que estamos dispuestos a aceptar) y si el ¨p-value \(< \alpha\), entonces rechazamos \(H_0\), de lo contrario, no rechazamos la \(H_0\).
Las pruebas para el modelo ingenuo:
# lag=h and fitdf=K
aug %>% features(.resid, box_pierce, lag=10, dof=0)# A tibble: 1 x 4
Symbol .model bp_stat bp_pvalue
<chr> <chr> <dbl> <dbl>
1 GOOG NAIVE(Close) 7.74 0.654aug %>% features(.resid, ljung_box, lag=10, dof=0)# A tibble: 1 x 4
Symbol .model lb_stat lb_pvalue
<chr> <chr> <dbl> <dbl>
1 GOOG NAIVE(Close) 7.91 0.637Estas pruebas nos indican que los residuos del modelo Naïve sí son ruido blanco (no están autocorrelacionados).
Para el modelo de la media:
google_2015 %>%
model(MEAN(Close)) %>% augment() %>%
features(.resid, box_pierce, lag = 10, dof = 0)# A tibble: 1 x 4
Symbol .model bp_stat bp_pvalue
<chr> <chr> <dbl> <dbl>
1 GOOG MEAN(Close) 2024. 0google_2015 %>%
model(MEAN(Close)) %>% augment() %>%
features(.resid, ljung_box, lag = 10, dof = 0)# A tibble: 1 x 4
Symbol .model lb_stat lb_pvalue
<chr> <chr> <dbl> <dbl>
1 GOOG MEAN(Close) 2083. 0En ambas pruebas, el p-value resulta ser muy alto, por lo que no podemos distinguir los residuales del ruido blanco.
Intervalos de predicción
Un intervalo de predicción se puede escribir como
\[\hat{y}_{T+h | T} \pm c \hat{\sigma}_{h}\] donde \(c\) es el porcentaje de cobertura de probabilidad. Normalmente utilizaremos 80% y 95%, pero se puede utilizar cualquier porcentaje.
La utilidad de los intervalos de predicción yace en el hecho de que nuestros pronósticos tienen cierta incertidumbre. Así, entre mayor sea el intervalo, menos preciso será nuestro pronóstico, y vice versa.
Un pronóstico puntual por si solo no sirve de gran cosa. Es necesario acompañarlo de su intervalo de predicción.
Intervalos de predicción de un paso
Cuando se realizan pronósticos de un paso (one-step forecast), la desviación estándar del pronóstico es prácticamente la misma que la desviación estándar de los residuos.
Intervalos de predicción de paso múltiple (multi-step)
Conforme se va aumentando el horizonte de pronostico, el intervalo de predicción tiende a aumentar. Entre más adelante en el tiempo queramos pronosticar, tendremos mayor incertidumbre (no es lo mismo querer predecir el tipo de cambio para mañana, que el de diciembre, p. ej.). Esto es, \(\sigma_h\) incrementa con \(h\). Entonces, requerimos estimaciones de \(\sigma_h\).
Para el caso del one-step forecast, ya vimos que podemos tomar la desviación estándar de los residuos como estimación de \(\sigma_h\). Para el caso multi-step, asumimos que los residuos no están autocorrelacionados y se requieren métodos de cálculo un poco más complejos.
Métodos de referencia
Si \(\hat{\sigma}\) es la desviación estándar de los residuos y \(\hat{\sigma}_{h}\) es la desviación estándar del pronóstico \(h\)-step, podemos calcular para cada método de pronóstico de referencia:
Pronósticos de media: \(\hat{\sigma}_{h}=\hat{\sigma} \sqrt{1+1 / T}\)
Pronósticos naïve: \(\hat{\sigma}_{h}=\hat{\sigma} \sqrt{h}\)
Pronósticos naïve estacionales: \(\hat{\sigma}_{h}=\hat{\sigma} \sqrt{k+1}\), donde \(k\) es la parte entera de \((h-1)/m\).
Pronósticos de drift: \(\hat{\sigma}_{h}=\hat{\sigma} \sqrt{h(1+h / T)}\).
Utilizando la paquetería fable, es muy sencillo obtener pronósticos y sus bandas.
google_2015 %>%
model(NAIVE(Close)) %>%
forecast(h = 10) %>%
hilo()Esto mismo se puede graficar, como lo hemos hecho anteriormente:
google_2015 %>%
model(NAIVE(Close)) %>%
forecast(h = 10) %>%
autoplot(google_2015)
Intervalos de predicción con residuales bootstrap
Cuando no es razonable asumir normalidad en los residuos, podemos aplicarles bootstraping, ya que esto solo asume la no autocorrelación.
Teníamos que los residuos se calculan \(e_{t}=y_{t}-\hat{y}_{t | t-1}\). Reescribiendo, tenemos que:
\[y_{t}=\hat{y}_{t | t-1}+e_{t}\] y podemos simular la siguiente observación de una serie de tiempo con:
\[y_{T+1}=\hat{y}_{T+1 | T}+e_{T+1}\]
donde \(\hat{y}_{T+1} | T\) es el pronóstico de un periodo (one-step) y \(e_{T+1}\) es el error futuro (que desconocemos). Ya que no están autocorelacionados los errores, y puesto que asumimos que los errores futuros serán similares a los históricos, podemos cambiar \(e_{T+1}\) al hacer un muestreo de los residuos. Podemos realizar el mismo proceso para \(y_{T+2}, y_{T+3}, \dots\).
Si realizamos esto varias veces, obtendremos muchos escenarios futuros posibles. Para ver algunos de ellos, utilizamos generate.
Dr. Strange simulando mediante bootstrap 14,000,605 escenarios posibles
fit <- google_2015 %>%
model(NAIVE(Close))
sim <- fit %>% generate(h = 30, times = 5, bootstrap = TRUE, seed = 123) # only works with dev version
simLo que hicimos fue generar 5 escenarios futuros posibles (times = 5) para los siguientes 30 días de trading (h = 30). Si graficamos esto, tenemos:
google_2015 %>%
ggplot(aes(x = day)) +
geom_line(aes(y = Close), size = 1) +
geom_line(aes(y = .sim, colour = as.factor(.rep)), data = sim, size = 1) +
ggtitle("Google closing stock price") +
guides(col = FALSE)Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> = "none")` instead.
Con esto, podemos obtener intervalos de predicción, al calcular los percentiles de los escenarios futuros. El resultado se llama intervalo de predicción bootstrapped. Esto se puede lograr fácilmente con forecast.
fc <- fit %>% forecast(h = 30, bootstrap = TRUE)
fcGraficando:
fc %>% autoplot(google_2015) +
ggtitle("Google closing stock price")
Tarea predicción de producción de ladrillos
aus_production %>%
autoplot(Bricks)Warning: Removed 20 row(s) containing missing values (geom_path).
aus_production %>%
autoplot(box_cox(Bricks, lambda = 0.4))Warning: Removed 20 row(s) containing missing values (geom_path).
aus_production %>%
select(Quarter, Bricks) %>%
drop_na() %>%
slice_tail(n = 8)
# Los datos de entrenamiento
bricks_train <- aus_production %>%
filter_index(. ~ "2003 Q2")# %>%
# mutate(log_bricks = log(Bricks)) # no deberían hacer la transformación antes del modelo
# Estimación de modelos
bricks_fit <- bricks_train %>%
model(
Media = MEAN(log(Bricks)),
Ingenuo = NAIVE(log(Bricks)),
`Ingenuo estacional` = SNAIVE(log(Bricks)),
Deriva = RW(log(Bricks) ~ drift())
)
bricks_aug <- augment(bricks_fit)
bricks_aug
graf_residuos <- function(modelo = "Media"){
bricks_fit %>%
select(modelo) %>%
gg_tsresiduals() +
ggtitle(paste("Diagnóstico de residuales para el modelo", modelo))
}
graf_residuos("Media")
graf_residuos("Ingenuo")Warning: Removed 1 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values
(geom_point).
Warning: Removed 1 rows containing non-finite values
(stat_bin).
graf_residuos("Ingenuo estacional")Warning: Removed 4 row(s) containing missing values (geom_path).
Warning: Removed 4 rows containing missing values
(geom_point).
Warning: Removed 4 rows containing non-finite values
(stat_bin).
graf_residuos("Deriva")Warning: Removed 1 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values
(geom_point).
Warning: Removed 1 rows containing non-finite values
(stat_bin).
bricks_fc <- bricks_fit %>%
forecast(h = "2 years")
bricks_fc %>%
autoplot(aus_production %>% select(Quarter, Bricks) %>% drop_na(), level = NULL)
bricks_fc %>%
autoplot(aus_production %>% select(Quarter, Bricks) %>% drop_na() %>% filter_index("1995 Q1" ~ .),
level = NULL)
bricks_fc %>%
filter(.model == "Ingenuo estacional") %>%
autoplot(aus_production %>% select(Quarter, Bricks) %>% drop_na() %>% filter_index("1995 Q1" ~ .))
Tarea
- Conseguir datos históricos sobre dos series de tiempo.
- Seguir los pasos del flujo de trabajo de pronóstico.
- Estimar los modelos de referencia (benchmark) que consideren adecuados para su serie.
- Realicen el diagnóstico de residuales e interpreten los resultados.
- Ejecuten un pronóstico (ustedes deciden el horizonte de pronóstico) y definan si utilizar el método bootstrap o no y justifiquen su decisión.
Pronósticos con transformaciones
¿Qué sucede cuando realizamos pronósticos de series a las que les hicimos alguna transformación? Por ejemplo, si los contratan para pronosticar las ventas de cubrebocas en la farmacia y ustedes hubieran realizado una transformación logarítmica de los datos, ¿cómo se realiza el pronóstico?
Cuando se realiza una transformación matemática, el pronóstico se hace con esa serie transformada, y posteriormente tenemos que darle reversa a la transformación (back-transformation), para obtener pronósticos en la escala original de la serie.
La transformación inversa de Box-Cox está dada por:
\[ y_{t}=\left\{\begin{array}{ll} \exp \left(w_{t}\right) & \text { si } \lambda=0 \\ \left(\lambda w_{t}+1\right)^{1 / \lambda} & \text { en otro caso } \end{array}\right. \]
La paquetería fable convenientemente realiza la transformación inversa en automático, cuando se especifica en la definición del modelo.
Intervalos de predicción con transformaciones
El intervalo de predicción de una serie transformada se calcula, primeramente, en la escala transformada, y posteriormente se hace la transformación inversa a la escala original. Hacer esto mantiene los porcentajes de cobertura de probabilidad originales, pero el resultado ya no es simétrico alrededor de la estimación puntual.
Es importante mencionar que las transformaciones tienen muy poco efecto en la estimación puntual, pero pueden llegar a tener un gran impacto en los intervalos de predicción.
Ajustes por sesgo
Un problema al realizar transformaciones matemáticas, como Box-Cox es que la estimaciones puntuales re transformadas ya no representan la media de la distribución de predicción, sino que representan ahora la mediana. En muchos casos puede no ser tan grave esto, pero en ocasiones el pronóstico promedio es requerido:
- P. ej., si quieren realizar el pronóstico de la venta de cubrebocas en las farmacias de la ZMG, para, al sumarlos, obtener el pronóstico de las ventas totales de cubrebocas en ZMG. La suma de las medias da como resultado el total, pero la suma de medianas no.
La transformación inversa de la media, para Box-Cox es:
\[ y_{t}=\left\{\begin{array}{ll} \exp \left(w_{t}\right)\left[1+\frac{\sigma_{h}^{2}}{2}\right] & \text { si } \lambda=0 \\ \left(\lambda w_{t}+1\right)^{1 / \lambda}\left[1+\frac{\sigma_{h}^{2}(1-\lambda)}{2\left(\lambda w_{t}+1\right)^{2}}\right] & \text { en otro caso } \end{array}\right. \] donde \(\sigma_{h}^{2}\) es la varianza del pronóstico en el horizonte-\(h\) en la escala transformada. Entre más grande sea la varianza, mayor será la diferencia entre la media y la mediana. A esto se le conoce como la estimación ajustada por sesgo.
Veamos un ejemplo sobre el pronóstico del precio promedio del huevo.
```r
eggs <- as_tsibble(fma::eggs)
eggs %>%
model(RW(log(value) ~ drift())) %>%
forecast(h=50) %>%
autoplot(eggs, level = 80,
point_forecast = lst(mean, median))
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La línea continua representa el pronóstico de la media (*ajustado por sesgo*), mientras que la línea punteada muestra el pronóstico de la mediana (*sin corrección por sesgo). `fable` en automático nos produce pronósticos ajustados por sesgo. Como se ve en el código, para generar pronósticos sesgados, tenemos que especificarlo mediante `point_forecast = lst(median)`.
# Pronósticos con descomposición
La descomposición de series de tiempo puede ser útil para producir pronósticos. Reescribiendo las fórmulas de descomposición aditiva y multiplicativa:
$$y_{t}=\hat{S}_{t}+\hat{A}_{t}$$
donde $\hat{A}_{t}$ es la serie desestacionalizada ( $\hat{A}_{t} = \hat{T}_{t} + \hat{R}_{t}$).
$$y_{t}=\hat{S}_{t}\hat{A}_{t}$$
con $\hat{A}_{t} = \hat{T}_{t} \hat{R}_{t}$.
Así, el pronóstico se realiza en dos pasos: un pronóstico para el componente estacional, y un pronóstico separado para la serie desestacionalizada. De hecho, el pronóstico del componente estacional es simplemente el método **naïve estacional**. Para los datos desestacionalizados, podemos utilizar cualquier modelo de pronóstico que veremos más adelante.
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<!-- rnb-chunk-begin -->
<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuYGBgclxudXNfcmV0YWlsX2VtcGxveW1lbnQgPC0gdXNfZW1wbG95bWVudCAlPiVcbiAgZmlsdGVyKHllYXIoTW9udGgpID49IDE5OTAsIFRpdGxlID09IFxcUmV0YWlsIFRyYWRlXFwpXG5cbnVzX3JldGFpbF9lbXBsb3ltZW50ICU+JSBcbmF1dG9wbG90KEVtcGxveWVkKVxuYGBgXG5gYGAifQ== -->
```r
```r
us_retail_employment <- us_employment %>%
filter(year(Month) >= 1990, Title == \Retail Trade\)
us_retail_employment %>%
autoplot(Employed)
<!-- rnb-source-end -->
<!-- rnb-plot-begin -->
<img src="data:image/png;base64,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" />
<!-- rnb-plot-end -->
<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuYGBgclxuXG5kY21wIDwtIHVzX3JldGFpbF9lbXBsb3ltZW50ICU+JVxuICBtb2RlbChTVEwoRW1wbG95ZWQgfiB0cmVuZCh3aW5kb3cgPSA3KSwgcm9idXN0PVRSVUUpKSAlPiVcbiAgY29tcG9uZW50cygpICU+JVxuICBzZWxlY3QoLS5tb2RlbClcblxuZGNtcFxuYGBgXG5gYGByXG5cbmRjbXAgJT4lXG4gIG1vZGVsKE5BSVZFKHNlYXNvbl9hZGp1c3QpKSAlPiVcbiAgZm9yZWNhc3QoKSAlPiVcbiAgYXV0b3Bsb3QoZGNtcCkgKyB5bGFiKFxcTmV3IG9yZGVycyBpbmRleFxcKSArXG4gIGdndGl0bGUoXFxQcm9uw4PCs3N0aWNvIG5hw4PCr3ZlIGRlIGxvcyBkYXRvcyBkZXNlc3RhY2lvbmFsaXphZG9zXFwpXG5gYGBcbmBgYCJ9 -->
```r
```r
dcmp <- us_retail_employment %>%
model(STL(Employed ~ trend(window = 7), robust=TRUE)) %>%
components() %>%
select(-.model)
dcmp
dcmp %>%
model(NAIVE(season_adjust)) %>%
forecast() %>%
autoplot(dcmp) + ylab(\New orders index\) +
ggtitle(\Pronóstico naïve de los datos desestacionalizados\)
<!-- rnb-source-end -->
<!-- rnb-frame-begin {"metadata":{"classes":["tbl_ts","tbl_df","tbl","data.frame"],"nrow":357,"ncol":7,"summary":{"A tsibble":"357 x 7 [1M]","Key":"Series_ID 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" />
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<!-- rnb-text-begin -->
A esta serie, podemos agregarle nuevamente la estacionalidad con la función `decomposition_model()`.
<!-- rnb-text-end -->
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<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuYGBgclxudXNfcmV0YWlsX2VtcGxveW1lbnQgJT4lXG4gIG1vZGVsKHN0bGYgPSBkZWNvbXBvc2l0aW9uX21vZGVsKFxuICAgICAgICAgICAgIFNUTChFbXBsb3llZCB+IHRyZW5kKHdpbmRvdyA9IDcpLCByb2J1c3QgPSBUUlVFKSxcbiAgICAgICAgICAgICBOQUlWRShzZWFzb25fYWRqdXN0KSxcbiAgICAgICAgICAgICBTTkFJVkUoc2Vhc29uX3llYXIpXG4gICkpICU+JVxuICBmb3JlY2FzdCgpICU+JVxuICBhdXRvcGxvdCh1c19yZXRhaWxfZW1wbG95bWVudClcbmBgYFxuYGBgIn0= -->
```r
```r
us_retail_employment %>%
model(stlf = decomposition_model(
STL(Employed ~ trend(window = 7), robust = TRUE),
NAIVE(season_adjust),
SNAIVE(season_year)
)) %>%
forecast() %>%
autoplot(us_retail_employment)
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<img 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/>
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# Evaluación del desempeño de los pronósticos
### Conjuntos de entrenamiento y prueba
Como hemos dicho, es muy importante separar nuestros datos en dos conjuntos: un conjunto de datos de **entrenamiento**, que son los que se utilizan para estimar el modelo, y un conjunto de datos de **prueba**, donde se evalúa el desempeño del pronóstico.
El tamaño de la prueba es, generalmente, del 20% del total de datos disponibles, aunque también puede depender del horizonte de pronóstico requerido. La prueba tiene que ser al menos tan grande como el horizonte de pronóstico más largo que se requiera (si se necesitan pronósticos para todo el siguiente año, la prueba tiene que ser del tamaño de al menos de todo el siguiente año).
**NOTA**: Es importante tener en cuenta lo siguiente:
* Un modelo que se ajusta muy bien a los datos de entrenamiento no necesariamente produce los mejores pronósticos.
* Podemos llegar a tener un ajuste perfecto del modelo a los datos, si aumentamos la cantidad de parámetros.
* Puede darse un efecto de sobre ajuste (mejor conocido como *over-fitting*) y esto es tan malo como tener un muy mal ajuste.
En alguna literatura o paqueterías pueden encontrar que al conjunto de datos de entrenamiento se les conozca como "*in-sample data*" y al conjunto de prueba "*out-of-sample data*".
### Funciones para segmentar las series de tiempo
Hemos visto que podemos utilizar la función `filter` para filtrar una base de datos o una serie de tiempo.
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```r
```r
aus_production
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</script>
</div>
<!-- rnb-frame-end -->
<!-- rnb-chunk-end -->
<!-- rnb-text-begin -->
Por ejemplo, tomando la producción en Australia, podemos filtrar para tener los datos a partir de 1995:
<!-- rnb-text-end -->
<!-- rnb-chunk-begin -->
<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuYGBgclxuYXVzX3Byb2R1Y3Rpb24gJT4lIGZpbHRlcih5ZWFyKFF1YXJ0ZXIpID49IDE5OTUpXG5gYGBcbmBgYCJ9 -->
```r
```r
aus_production %>% filter(year(Quarter) >= 1995)
<!-- rnb-source-end -->
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Q3","2":"398","3":"6293","4":"379","5":"1706","6":"46326","7":"184"},{"1":"1996 Q4","2":"507","3":"5575","4":"369","5":"1699","6":"43346","7":"146"},{"1":"1997 Q1","2":"432","3":"4802","4":"330","5":"1511","6":"43938","7":"135"},{"1":"1997 Q2","2":"398","3":"5523","4":"390","5":"1785","6":"45828","7":"171"},{"1":"1997 Q3","2":"406","3":"5708","4":"416","5":"1826","6":"48448","7":"183"},{"1":"1997 Q4","2":"526","3":"4821","4":"383","5":"1830","6":"46096","7":"151"},{"1":"1998 Q1","2":"428","3":"4919","4":"339","5":"1719","6":"46198","7":"141"},{"1":"1998 Q2","2":"397","3":"5809","4":"394","5":"1861","6":"47237","7":"174"},{"1":"1998 Q3","2":"403","3":"5904","4":"412","5":"1956","6":"49041","7":"191"},{"1":"1998 Q4","2":"517","3":"4555","4":"420","5":"2067","6":"46192","7":"157"},{"1":"1999 Q1","2":"435","3":"5198","4":"376","5":"1737","6":"48089","7":"145"},{"1":"1999 Q2","2":"383","3":"5388","4":"401","5":"1944","6":"48227","7":"182"},{"1":"1999 Q3","2":"424","3":"5142","4":"430","5":"2005","6":"49784","7":"198"},{"1":"1999 Q4","2":"521","3":"5517","4":"417","5":"2027","6":"47588","7":"165"},{"1":"2000 Q1","2":"421","3":"5169","4":"416","5":"1835","6":"49320","7":"164"},{"1":"2000 Q2","2":"402","3":"4860","4":"447","5":"2070","6":"50670","7":"199"},{"1":"2000 Q3","2":"414","3":"5185","4":"421","5":"1898","6":"52623","7":"213"},{"1":"2000 Q4","2":"500","3":"4763","4":"379","5":"1652","6":"49350","7":"173"},{"1":"2001 Q1","2":"451","3":"4217","4":"304","5":"1554","6":"51658","7":"177"},{"1":"2001 Q2","2":"380","3":"4959","4":"337","5":"1717","6":"51103","7":"205"},{"1":"2001 Q3","2":"416","3":"5196","4":"385","5":"1679","6":"52226","7":"218"},{"1":"2001 Q4","2":"492","3":"4522","4":"381","5":"1836","6":"50778","7":"185"},{"1":"2002 Q1","2":"428","3":"3843","4":"345","5":"1729","6":"50639","7":"166"},{"1":"2002 Q2","2":"408","3":"4806","4":"405","5":"1992","6":"51486","7":"204"},{"1":"2002 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Q3","2":"408","3":"NA","4":"NA","5":"2340","6":"56043","7":"221"},{"1":"2005 Q4","2":"482","3":"NA","4":"NA","5":"2265","6":"54992","7":"180"},{"1":"2006 Q1","2":"438","3":"NA","4":"NA","5":"2027","6":"57112","7":"171"},{"1":"2006 Q2","2":"386","3":"NA","4":"NA","5":"2278","6":"57157","7":"224"},{"1":"2006 Q3","2":"405","3":"NA","4":"NA","5":"2427","6":"58400","7":"233"},{"1":"2006 Q4","2":"491","3":"NA","4":"NA","5":"2451","6":"56249","7":"192"},{"1":"2007 Q1","2":"427","3":"NA","4":"NA","5":"2140","6":"56244","7":"187"},{"1":"2007 Q2","2":"383","3":"NA","4":"NA","5":"2362","6":"55036","7":"234"},{"1":"2007 Q3","2":"394","3":"NA","4":"NA","5":"2536","6":"59806","7":"245"},{"1":"2007 Q4","2":"473","3":"NA","4":"NA","5":"2562","6":"56411","7":"205"},{"1":"2008 Q1","2":"420","3":"NA","4":"NA","5":"2183","6":"59118","7":"194"},{"1":"2008 Q2","2":"390","3":"NA","4":"NA","5":"2558","6":"56660","7":"229"},{"1":"2008 Q3","2":"410","3":"NA","4":"NA","5":"2612","6":"64067","7":"249"},{"1":"2008 Q4","2":"488","3":"NA","4":"NA","5":"2373","6":"59045","7":"203"},{"1":"2009 Q1","2":"415","3":"NA","4":"NA","5":"1963","6":"58368","7":"196"},{"1":"2009 Q2","2":"398","3":"NA","4":"NA","5":"2160","6":"57471","7":"238"},{"1":"2009 Q3","2":"419","3":"NA","4":"NA","5":"2325","6":"58394","7":"252"},{"1":"2009 Q4","2":"488","3":"NA","4":"NA","5":"2273","6":"57336","7":"210"},{"1":"2010 Q1","2":"414","3":"NA","4":"NA","5":"1904","6":"58309","7":"205"},{"1":"2010 Q2","2":"374","3":"NA","4":"NA","5":"2401","6":"58041","7":"236"}],"options":{"columns":{"min":{},"max":[10],"total":[7]},"rows":{"min":[10],"max":[10],"total":[62]},"pages":{}}}
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<!-- rnb-frame-end -->
<!-- rnb-chunk-end -->
<!-- rnb-text-begin -->
o para obtener los datos de una cierta estación:
<!-- rnb-text-end -->
<!-- rnb-chunk-begin -->
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```r
```r
aus_production %>% filter(quarter(Quarter) == 1)
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{"columns":[{"label":["Quarter"],"name":[1],"type":["S3: yearquarter"],"align":["right"]},{"label":["Beer"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["Tobacco"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["Bricks"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["Cement"],"name":[5],"type":["dbl"],"align":["right"]},{"label":["Electricity"],"name":[6],"type":["dbl"],"align":["right"]},{"label":["Gas"],"name":[7],"type":["dbl"],"align":["right"]}],"data":[{"1":"1956 Q1","2":"284","3":"5225","4":"189","5":"465","6":"3923","7":"5"},{"1":"1957 Q1","2":"262","3":"5577","4":"187","5":"529","6":"4339","7":"5"},{"1":"1958 Q1","2":"272","3":"5758","4":"199","5":"554","6":"4608","7":"5"},{"1":"1959 Q1","2":"261","3":"5478","4":"208","5":"573","6":"4883","7":"5"},{"1":"1960 Q1","2":"286","3":"5980","4":"242","5":"621","6":"5387","7":"6"},{"1":"1961 Q1","2":"295","3":"6022","4":"241","5":"688","6":"5709","7":"6"},{"1":"1962 Q1","2":"279","3":"6072","4":"229","5":"637","6":"6098","7":"6"},{"1":"1963 Q1","2":"294","3":"6051","4":"231","5":"674","6":"6707","7":"6"},{"1":"1964 Q1","2":"313","3":"5890","4":"293","5":"797","6":"7469","7":"6"},{"1":"1965 Q1","2":"331","3":"6201","4":"309","5":"902","6":"8170","7":"6"},{"1":"1966 Q1","2":"335","3":"5522","4":"302","5":"849","6":"8650","7":"6"},{"1":"1967 Q1","2":"353","3":"5801","4":"306","5":"836","6":"9490","7":"6"},{"1":"1968 Q1","2":"393","3":"6255","4":"341","5":"892","6":"10087","7":"6"},{"1":"1969 Q1","2":"383","3":"6641","4":"383","5":"948","6":"11136","7":"7"},{"1":"1970 Q1","2":"387","3":"6807","4":"386","5":"1049","6":"12328","7":"12"},{"1":"1971 Q1","2":"410","3":"7255","4":"385","5":"1058","6":"13299","7":"19"},{"1":"1972 Q1","2":"419","3":"6500","4":"399","5":"1070","6":"13838","7":"24"},{"1":"1973 Q1","2":"458","3":"7168","4":"452","5":"1148","6":"15024","7":"34"},{"1":"1974 Q1","2":"465","3":"6823","4":"478","5":"1184","6":"16249","7":"38"},{"1":"1975 Q1","2":"500","3":"6971","4":"340","5":"1121","6":"16692","7":"39"},{"1":"1976 Q1","2":"510","3":"6687","4":"424","5":"1109","6":"18023","7":"44"},{"1":"1977 Q1","2":"486","3":"7156","4":"457","5":"1101","6":"19031","7":"49"},{"1":"1978 Q1","2":"515","3":"7219","4":"421","5":"1122","6":"20084","7":"56"},{"1":"1979 Q1","2":"503","3":"7084","4":"458","5":"1205","6":"21323","7":"59"},{"1":"1980 Q1","2":"513","3":"6794","4":"513","5":"1232","6":"22604","7":"68"},{"1":"1981 Q1","2":"548","3":"6437","4":"519","5":"1282","6":"23639","7":"88"},{"1":"1982 Q1","2":"493","3":"6697","4":"501","5":"1337","6":"24675","7":"93"},{"1":"1983 Q1","2":"475","3":"6559","4":"303","5":"1086","6":"24687","7":"92"},{"1":"1984 Q1","2":"453","3":"6631","4":"400","5":"1166","6":"26641","7":"97"},{"1":"1985 Q1","2":"464","3":"6033","4":"447","5":"1251","6":"28982","7":"103"},{"1":"1986 Q1","2":"459","3":"5733","4":"442","5":"1375","6":"29506","7":"98"},{"1":"1987 Q1","2":"481","3":"5765","4":"414","5":"1332","6":"31015","7":"106"},{"1":"1988 Q1","2":"474","3":"6034","4":"428","5":"1418","6":"33142","7":"116"},{"1":"1989 Q1","2":"467","3":"6051","4":"510","5":"1621","6":"36905","7":"117"},{"1":"1990 Q1","2":"485","3":"6562","4":"458","5":"1565","6":"37016","7":"125"},{"1":"1991 Q1","2":"464","3":"6202","4":"372","5":"1343","6":"37554","7":"111"},{"1":"1992 Q1","2":"443","3":"5777","4":"383","5":"1289","6":"38332","7":"117"},{"1":"1993 Q1","2":"433","3":"5724","4":"394","5":"1450","6":"39460","7":"116"},{"1":"1994 Q1","2":"449","3":"5311","4":"421","5":"1468","6":"40162","7":"127"},{"1":"1995 Q1","2":"426","3":"4714","4":"430","5":"1626","6":"41768","7":"131"},{"1":"1996 Q1","2":"409","3":"4275","4":"310","5":"1526","6":"43661","7":"133"},{"1":"1997 Q1","2":"432","3":"4802","4":"330","5":"1511","6":"43938","7":"135"},{"1":"1998 Q1","2":"428","3":"4919","4":"339","5":"1719","6":"46198","7":"141"},{"1":"1999 Q1","2":"435","3":"5198","4":"376","5":"1737","6":"48089","7":"145"},{"1":"2000 Q1","2":"421","3":"5169","4":"416","5":"1835","6":"49320","7":"164"},{"1":"2001 Q1","2":"451","3":"4217","4":"304","5":"1554","6":"51658","7":"177"},{"1":"2002 Q1","2":"428","3":"3843","4":"345","5":"1729","6":"50639","7":"166"},{"1":"2003 Q1","2":"435","3":"4362","4":"387","5":"1831","6":"51415","7":"172"},{"1":"2004 Q1","2":"435","3":"3974","4":"409","5":"1963","6":"54561","7":"173"},{"1":"2005 Q1","2":"416","3":"NA","4":"355","5":"1980","6":"55035","7":"170"},{"1":"2006 Q1","2":"438","3":"NA","4":"NA","5":"2027","6":"57112","7":"171"},{"1":"2007 Q1","2":"427","3":"NA","4":"NA","5":"2140","6":"56244","7":"187"},{"1":"2008 Q1","2":"420","3":"NA","4":"NA","5":"2183","6":"59118","7":"194"},{"1":"2009 Q1","2":"415","3":"NA","4":"NA","5":"1963","6":"58368","7":"196"},{"1":"2010 Q1","2":"414","3":"NA","4":"NA","5":"1904","6":"58309","7":"205"}],"options":{"columns":{"min":{},"max":[10],"total":[7]},"rows":{"min":[10],"max":[10],"total":[55]},"pages":{}}}
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<!-- rnb-frame-end -->
<!-- rnb-chunk-end -->
<!-- rnb-text-begin -->
Otra función útil para filtrar o segmentar series de tiempo es `slice()`, que utiliza el índice para filtrar los datos.
<!-- rnb-text-end -->
<!-- rnb-chunk-begin -->
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```r
```r
aus_production %>%
slice(n()-19:0)
<!-- rnb-source-end -->
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<!-- rnb-text-begin -->
Esto filtra los últimos 20 datos (5 años).
A partir de `dplyr 1.0.0`, existen algunas variantes de `slice()` que pueden sernos bastante útiles, como `slice_head()`, `slice_tail()`, `slice_sample()`, `slice_max()`, etc.
Podemos reescribir el código de arriba para hacerlo más explícito:
<!-- rnb-text-end -->
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```r
```r
aus_production %>%
slice_tail(n = 20)
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<!-- rnb-chunk-end -->
<!-- rnb-text-begin -->
Podemos usar `slice` para datos agrupados:
<!-- rnb-text-end -->
<!-- rnb-chunk-begin -->
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```r
```r
aus_retail %>%
group_by(State, Industry) %>%
slice(1:12)
<!-- rnb-source-end -->
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<!-- rnb-text-begin -->
Otra función es `top_n`, la cual nos permite obtener las `n` observaciones más extremas.
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<!-- rnb-chunk-begin -->
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```r
```r
gafa_stock %>%
group_by(Symbol) %>%
top_n(1, Close)
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## Errores de pronóstico
El error de pronóstico es la diferencia entre el valor real ocurrido y el dato pronosticado.
$$e_{T+h}=y_{T+h}-\hat{y}_{T+h | T}$$
Los errores de pronóstico son distintos de los residuales en dos formas:
1. Los residuales se calculan con los datos de *entrenamiento*, mientras que los errores de pronóstico se calculan con los de *prueba*.
2. Los residuos se calculan mediante pronósticos de un paso (one-step), donde los errores de pronóstico pueden ser multi-step.
Hay muchos tipos de cálculo del error general de un modelo de pronóstico.
### Errores dependientes de la escala de los datos
Los errores de pronóstico están medidos en la misma escala de los datos originales. Hay ciertos tipos de error que solo se basan en
$e_{T}$, por lo que también dependen de la escala y no pueden ser utilizados para comparar el desempeño con otra serie de tiempo que tenga otras unidades.
Los dos más utilizados en este rubro son el **MAE** y el **RMSE**.
$$
\begin{aligned}
&\text { Mean absolute error: } \mathrm{MAE}=\operatorname{mean}\left(\left|e_{t}\right|\right)\\
&\text { Root mean squared error: } \operatorname{RMSE}=\sqrt{\operatorname{mean}\left(e_{t}^{2}\right)}
\end{aligned}
$$
El MAE es muy utilizado debido a su facilidad de cómputo y de interpretación. Un método de pronóstico que minimiza el MAE nos dará pronósticos de la mediana de la distribución. Pronósticos que minimizan el RMSE obtienen pronósticos de la media, por lo que este método también es muy utilizado, a pesar de ser más pesado computacionalmente y complicado de interpretar.
### Errores porcentuales
Los errores porcentuales, al ser un porcentaje, no tienen unidades y son utilizados para comparar el desempeño de pronósticos de distintos conjuntos de datos. El más utilizado es el MAPE (error absoluto promedio porcentual). Si definimos al error porcentual como $p_{t}=100 e_{t} / y_{t}$
Mean absolute percentage error: **MAPE** $=\operatorname{mean}\left(\left|p_{t}\right|\right)$
La desventaja con estos errores es que se indeterminana o vuelven infinitos con valores de $y_t = 0$. Para ello, se definió el MAPE simétrico:
$$\operatorname{sMAPE}=\operatorname{mean}\left(200\left|y_{t}-\hat{y}_{t}\right| /\left(y_{t}+\hat{y}_{t}\right)\right)$$
Aunque este método no se recomienda tanto utilizarlo en la práctica.
### Errores escalados
Una alternativa a los errores porcentuales cuando se quieren comparar los pronósticos de distintas series de tiempo so los **errores escalados**, propuestos por Hyndman & Koehler. En éstos, se escalan los errores con base en el MAE de entrenamiento de utilizar un modelo Naïve o Seasonal Naïve, dependiendo de si los datos tienen estacionalidad.
Para series no estacionales, los errores escalados serían:
$$
q_{j}=\frac{e_{j}}{\frac{1}{T-1} \sum_{t=2}^{T}\left|y_{t}-y_{t-1}\right|},
$$
mientras que para series estacionales, los errores escalados estarían dados por:
$$
q_{j}=\frac{e_{j}}{\frac{1}{T-m} \sum_{t=m+1}^{T}\left|y_{t}-y_{t-m}\right|}.
$$
De esta forma podemos obtener el **Error Medio Absoluto Escalado (MASE)** con
$$
\operatorname{MASE} = \operatorname{mean}(|q_j|).
$$
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```r
```r
recent_production <- aus_production %>% filter(year(Quarter) >= 1992)
beer_train <- recent_production %>% filter(year(Quarter) <= 2007)
beer_fit <- beer_train %>%
model(
Mean = MEAN(Beer),
`Naïve` = NAIVE(Beer),
`Seasonal naïve` = SNAIVE(Beer),
Drift = RW(Beer ~ drift())
)
beer_fc <- beer_fit %>%
forecast(h = 10)
beer_fc %>%
autoplot(recent_production, level = NULL) +
xlab(\Year\) + ylab(\Megalitres\) +
ggtitle(\Forecasts for quarterly beer production\) +
guides(colour=guide_legend(title=\Forecast\))
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" />
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La función `accuracy()` nos permite calcular varias de estas métricas de error de manera muy sencilla. Incluso, podemos calcularlas para el ajuste del modelo a los datos de entrenamiento, o calcular el error de pronóstico. Todo depende de los datos que le pasemos a la función.
Si utilizamos la tabla de modelos (`mable`), obtendremos las métricas de error para el ajuste del modelo a sus datos de entrenamiento:
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```r
```r
accuracy(beer_fit)
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Las métricas de error para el ajuste a los datos de entrenamiento nos pueden servir para decidir qué modelos utilizar para pronosticar.
En cambio, si utilizamos la tabla de pronósticos (`fable`), junto con los datos de validación (o datos de prueba), obtendremos los errores de pronóstico:
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```r
```r
beer_fc %>%
accuracy(recent_production)```
Las métricas de error sobre los datos de prueba nos dan una mayor claridad de cuáles modelos parecen producir los mejores pronósticos.
Una vez decidido esto, se recalculan los modelos utilizando toda la información histórica (es decir, sin separar en conjuntos de entrenamiento y prueba), y se producen los pronósticos reales a presentar.
---
title: "Las bases de los pronósticos"
author: "Pablo Benavides-Herrera"
date: 2020-04-06
output:
  html_notebook:
    toc: yes
    toc_float: yes
    theme: united
    highlight: tango
  github_document:
    toc: yes
    dev: jpeg
always_allow_html: TRUE
---

![](../images/tsibble.png){width=30%} 

![](../images/fable.png){width=30%}
![](../images/feasts.png){width=30%} 


Las paqueterías a usar en este documento:

```{r, message=FALSE}
library("easypackages")
packages("tidyverse","fpp3", "tsibble", "feasts","fable", "patchwork")
```


# Un flujo de trabajo limpio de pronóstico

El flujo de trabajo, cuando se va a realizar un proceso, se puede dividir en pasos.

1. Preparación de los datos (limpieza)
2. Gráfica de los datos (visualización)
3. Definición del modelo (especificación)
4. Entrenamiento del modelo (estimación)
5. Revisar el desempeño del modelo (evaluación)
6. Producir pronósticos

## 1. Preparación de los datos (limpieza)

Esto siempre es lo primero que se hace y puede ir desde simplemente cargar los datos en **R**, hasta limpiezas más detalladas, como identificar valores omitidos, `NA`, filtrado de la serie, etc. Para esto, utilizamos varias funciones de las paqueterías `tsibble` y `tidyverse`, que simplifican mucho estas tareas.

## 2. Gráfica de los datos (visualización)

Continuaremos trabajando con los datos de `global_economy` para ejemplificar esto.

```{r visualización}
global_economy %>%
  filter(Country == "Sweden") %>%
  autoplot(GDP) +
    ggtitle("PIB de Suecia") + ylab("$US billions")
```

## 3. Definición del modelo (especificación)

Antes de ajustar un modelo a los datos, se debe **describir** el modelo. Existen muchos tipos de modelos de pronóstico distintos, y es muy importante que escojamos el más apropiado, para obtener buenos pronósticos.

Los modelos en **R**, como vimos con la regresión lineal (`lm(y ~ x1 + x2 + ... + xn)`), se especifican en formato de fórmula ( `y ~ x`), siendo la `y` la variable dependiente (o variable a explicar), y la o las `x` las variables independientes (o variables explicativas, regresoras, etc.).

Podemos tomar, p. ej., un modelo lineal de series de tiempo, `TSLM`, modela los datos que se le incluyan mediante una tendencia lineal.

```{r definición, eval=FALSE}
TSLM(GDP  ~ trend())
```


## 4. Entrenamiento del modelo (estimación)

Una vez que se especificó el modelo, lo que sigue es entrenar al modelo.

Entrenar un modelo significa pasarle los datos para que, estadísticamente, encuentre los parámetros que realizan el mejor ajuste posible.

Siguiendo con el ejemplo del modelo lineal:

```{r entrenamiento, warning=FALSE}
fit <- global_economy %>%
  model(Modelo_tendencia = TSLM(GDP ~ trend()))
```

Con esto se ajustó un modelo lineal y el objeto resultante es un `mable` (model table).

```{r tabla entrenamiento, paged.print=FALSE}
fit
```

## 5. Revisar el desempeño del modelo (evaluación)

Ya teniendo el modelo entrenado, debemos revisar el *performance* en los datos reales. Esto es, ¿qué tan bien se ajusta el modelo a los datos?, en caso de estar indecisos entre varios modelos, ¿cuál escogemos como el mejor y por qué?

## 6. Producir pronósticos

Cuando ya evaluamos que el modelo ajustado se encuentra dentro de los parámetros deseados, podemos proceder a realizar los pronósticos. En .**R**, podemos usar el comando `forecast()`, en el cual debemos especificar el número de periodos a pronosticar. Por ejemplo, par pronosticar los siguientes 12 meses, escribiríamos `h = 12`. También podemos usar lenguaje ordinario (en inglés), `h = "1 year"`.

```{r pronóstico, warning=FALSE, paged.print=FALSE}
fcst <- fit %>% forecast(h = "3 years")
fcst
```

El resultado es una `fable` (forecasting table) o tabla de pronósticos. El pronóstico se puede graficar fácilmente junto con los datos reales, usando `autoplot()`.

```{r gráfica pronóstico, warning=FALSE}
fcst %>% 
  filter(Country=="Sweden") %>%
  autoplot(global_economy) +
    ggtitle("PIB de Suecia") + ylab("$US billions")
```

# Métodos sencillos de pronóstico { .tabset}

Utilizaremos de *benchmark* a lo largo del curso estos métodos básicos de pronóstico. En ocasiones, a pesar de su sencillez, pueden llegar a ser muy útiles.

Utilizaremos los datos de producción de ladrillos para esta sección.

```{r bricks}
bricks <- aus_production %>% filter_index("1970" ~ "2004")
bricks
bricks %>% autoplot(Bricks)
```



## Método del promedio (media)

En este pronóstico, las predicciones de todos los valores futuros son la media de los datos históricos.

$$
\hat{y}_{T+h | T}=\bar{y}=\left(y_{1}+\cdots+y_{T}\right) / T
$$

```{r warning=FALSE, paged.print=FALSE}
bricks %>% model(MEAN(Bricks))
```

## Método ingenuo (Naïve method)

Aquí, lo que se hace es que se toma el último valor como el pronóstico para todos los valores futuros.

$$
\hat{y}_{T+h | T}=y_{T}
$$
Dado que un pronóstico ingenuo es óptimo cuando se tienen datos que siguen una *caminata aleatoria*, a estos pronósticos se les conoce como **pronósticos de caminata aleatoria**.

```{r paged.print=FALSE}
bricks %>% model(NAIVE(Bricks),
                 RW(Bricks)) # hace exactamente lo mismo que NAIVE()
```

```{r}
tidyquant::tq_get("AAPL") %>% 
  timetk::plot_time_series(.date_var = date,.value = close, .smooth = FALSE)
```

## Método ingenuo estacional (seasonal Naïve)

Un método similar es el ingenuo estacional. Lo que cambia con el anterior es que se agrega un componente para lidiar con datos altamente estacionales.

$$
\hat{y}_{T+h | T}=y_{T+h-m(k+1)}
$$
```{r message=FALSE, paged.print=FALSE}
bricks %>% model(SNAIVE(Bricks))
```



```{r}
vic_elec %>% 
  autoplot(Demand) %>% 
  plotly::ggplotly()
```

```{r}
vic_elec %>% 
  model(SNAIVE(Demand)) %>% 
  forecast(h = "1 day") %>% 
  autoplot(vic_elec %>% filter_index("2014-12-30" ~ .))
```


## Método del drift (deriva)

Este método es una variación del método ingenuo, que permite que el pronóstico aumente o disminuya en el tiempo. El aumento del cambio es el cambio promedio en los datos históricos.

$$
\hat{y}_{T+h | T}=y_{T}+\frac{h}{T-1} \sum_{t=2}^{T}\left(y_{t}-y_{t-1}\right)=y_{T}+h\left(\frac{y_{T}-y_{1}}{T-1}\right)
$$

```{r paged.print=FALSE}
bricks %>% model(
  RW(Bricks ~ drift())
)
```

Esto es lo mismo que trazar una línea recta que conecte el primer y último punto en los datos históricos y continuar la recta hacia adelante.

# {.unlisted .unnumbered}


```{r beer_fit y fcst}
# Set training data from 1992 to 2006
train <- aus_production %>% filter_index("1992 Q1" ~ "2006 Q4")
# Fit the models
beer_fit <- train %>%
  model(
    Mean             = MEAN(Beer),
    `Naïve`          = NAIVE(Beer),
    `Seasonal naïve` = SNAIVE(Beer),
    Drift            = RW(Beer ~ drift())
  )
# Generate forecasts for 14 quarters
beer_fc <- beer_fit %>% forecast(h= 14)

# Plot forecasts against actual values
beer_fc %>%
  autoplot(filter_index(aus_production, "1992 Q1" ~ .), level = NULL) +
  ggtitle("Forecasts for quarterly beer production") +
  xlab("Year") + ylab("Megalitres") +
  guides(colour=guide_legend(title="Forecast")) +
  geom_vline(xintercept = as.Date("2007-01-01"), color = "firebrick",
             linetype = "dashed") +
  annotate("label", x = c(as.Date("2003-01-01"),as.Date("2009-01-01")),
           y = 550, label = c("Train set", "Test set"),
           color = c("black","blue"))
```

```{r}
beer_fc %>%
  filter(.model == "Seasonal naïve") %>% 
  autoplot(filter_index(aus_production, "1992 Q1" ~ .)) +
  ggtitle("Seasonal naïve forecast for quarterly beer production") +
  xlab("Year") + ylab("Megalitres") +
  guides(colour=guide_legend(title="Forecast")) +
  geom_vline(xintercept = as.Date("2007-01-01"), color = "firebrick",
             linetype = "dashed") +
  annotate("label", x = c(as.Date("2003-01-01"),as.Date("2009-01-01")),
           y = 550, label = c("Train set", "Test set"),
           color = c("black","blue"))
```



Otro ejemplo:

```{r plot google fcsts}
gafa_stock %>% distinct(Symbol)
# Re-index based on trading days
google_stock <- gafa_stock %>%
  filter(Symbol == "GOOG") %>%
  mutate(day = row_number()) %>%
  update_tsibble(index = day, regular = TRUE)

# Filter the year of interest
google_2015 <- google_stock %>% filter(year(Date) == 2015)
# Fit the models
google_fit <- google_2015 %>%
  model(
    Mean    = MEAN(Close),
    `Naïve` = NAIVE(Close),
    Drift   = NAIVE(Close ~ drift()),
    SNAIVE  = SNAIVE(Close)
  )

google_fit
# Produce forecasts for the 19 trading days in January 2015
google_fc <- google_fit %>% forecast(h = 19)

google_fc
# A better way using a tsibble to determine the forecast horizons
google_jan_2016 <- google_stock %>%
  filter(yearmonth(Date) == yearmonth("2016 Jan"))
google_fc <- google_fit %>% forecast(google_jan_2016)
# Plot the forecasts
google_fc %>%
  autoplot(google_2015, level = NULL) +
    autolayer(google_jan_2016, Close, color='black') +
    ggtitle("Google stock (daily ending 31 Dec 2015)") +
    xlab("Day") + ylab("Closing Price (US$)") +
    guides(colour=guide_legend(title="Forecast"))
```

```{r, fig.height=9, fig.width=7}
p1 <- google_fc %>%
  filter(.model == "Drift") %>% 
  autoplot(google_2015) +
    autolayer(google_jan_2016, Close, color='black') +
    ggtitle("Google stock (daily ending 31 Dec 2015)") +
    xlab("Day") + ylab("Closing Price (US$)") +
    guides(colour=guide_legend(title="Forecast"))

p2 <- google_fc %>%
  filter(.model == "Naïve") %>% 
  autoplot(google_2015) +
    autolayer(google_jan_2016, Close, color='black') +
    ggtitle("Google stock (daily ending 31 Dec 2015)") +
    xlab("Day") + ylab("Closing Price (US$)") +
    guides(colour=guide_legend(title="Forecast"))

p1 / p2
```



# Valores ajustados *(fitted)* y residuales


Cada observación en una serie de tiempo puede ser pronosticada utilizando los datos históricos previos. A estos se les conoce como valores ajustados (o *fitted*), $\hat{y}_t$.

Los residuales en un modelo de series de tiempo es la información que el modelo no logró capturar. Esto es, es la diferencia entre los valores reales y los valores ajustados.

$$
e_{t}=y_{t}-\hat{y}_{t}
$$

En **R**, podemos obtener los valores ajustados y los residuales con la función `augment()`. Recordando, habíamos ajustado tres modelos distintos, que guardamos en la variable `beer_fit`.

```{r}
?augment
```


```{r augment}
augment(beer_fit)
```

Es muy importante analizar los residuos para determinar si nuestros modelos están bien ajustados. Si logramos detectar patrones en los residuales, puede ser indicio de que el modelo puede mejorarse.

# Diagnóstico de residuales

Un buen modelo de pronóstico va a producir residuales con las siguientes características:

1. **No están autocorrelacionados**. Si se detectan correlaciones entre residuos, todavía hay información útil que se debe modelar.

2. **La media de los residuos es cero**. Si la media es distinta de cero, entonces el pronóstico está sesgado.

**Nota:** El hecho de que un pronóstico cumpla esto, no quiere decir que sea el mejor pronóstico que podamos hacer. Próximamente revisaremos qué otras medidas podemos evaluar para determinar cuál es el mejor pronóstico.

Existen dos características adicionales que son útiles, mas no necesarias, para los residuos de un pronóstico:

3. Los residuos tienen una varianza constante.

4. Los residuos se distribuyen de manera normal.

Las transformaciones de Box-Cox pueden ayudar, en algunos casos a lograr cumplir estas características.

Continuemos con el ejemplo del pronóstico del precio de la acción de Google. En muchas ocasiones, el mejor pronóstico para los precios de mercados bursátiles e índices suele ser el realiado mediante el método Naïve.


```{r google plot}
google_2015 %>% autoplot(Close) +
  xlab("Day") + ylab("Closing Price (US$)") +
  ggtitle("Google Stock in 2015")
```

Aplicaremos solo el método Naïve para pronosticar el precio futuro de la acción de Google.

```{r residuales google}
aug <- google_2015 %>% 
  model(NAIVE(Close)) %>% 
  augment()

aug

# aug %>% 
#   features(.resid, mean(.,na.rm = TRUE))

aug %>% pull(.resid) %>% mean(na.rm = TRUE) 

aug %>% autoplot(.resid) + xlab("Día") + ylab("") +
  ggtitle("Residuales del método naïve")
```

De la gráfica de los residuales podemos observar que la media parece estar muy cercana al cero y que la variación parece invariante en el tiempo, a excepción de un outlier.

```{r resids hist, warning=FALSE}
aug %>%
  ggplot(aes(x = .resid)) +
  geom_histogram() +
  ggtitle("Histograma de los residuales")
```
Del histograma, podemos ver que los residuales parecen distribuirse como una normal, pero con una cola más grande.

```{r acf resids}
aug %>% ACF(.resid)

aug %>% ACF(.resid) %>% autoplot() + ggtitle("ACF of residuals")
```
```{r}
google_2015 %>% 
  model(NAIVE(Close)) %>% 
  gg_tsresiduals()
```

```{r}
aus_production %>% 
  filter_index("1992" ~ .) %>%
  model(SNAIVE(Beer)) %>% 
  gg_tsresiduals()

aus_production %>% 
  filter_index("1992" ~ .) %>%
  gg_tsdisplay(Beer)
```



La función de autocorrelación muestra que los residuos no están autocorrelacionados.

Así, el método naïve parece generar pronósticos que capturan toda la información relevante de la serie y, por lo tanto, que satisfacen todas las características necesarias. Por consecuencia, los pronósticos derivados de este método pueden ser bastante buenos, pero los intervalos de predicción pudieran ser imprecisos.

Podemos obtener estas mismas gráficas con un solo comando, `gg_tsresiduals()`.

```{r gg_tsresiduals naive, warning=FALSE}
google_2015 %>% 
  model(NAIVE(Close)) %>% 
  gg_tsresiduals() + 
  ggtitle("Diagnóstico de residuales para el modelo Naïve")
```
Como vimos antes, el método Naïve es óptimo para este tipo de series. ¿Qué hubiera sucedido si ajustáramos otro modelo? Probemos con el método de la media:
```{r gg_tsresiduals mean, warning=FALSE}
google_2015 %>% 
  model(MEAN(Close)) %>% 
  augment() %>% 
  pull(.resid) %>% 
  mean(na.rm = TRUE)

google_2015 %>% 
  model(MEAN(Close)) %>% 
  gg_tsresiduals() + 
  ggtitle("Diagnóstico de residuales para el modelo de Media")
```

Observamos que estas gráficas tienen un comportamiento muy distinto al Naïve:

* En la gráfica de los residuales vemos que se distingue claramente un patrón. De hecho, es el mismo patrón exactamente que siguen los datos originales, restándoles su media.

* La función de autocorrelación tiene un comportamiento típico de una caminata aleatoria. Por lo tanto, las autocorrelaciones son significativas.

* El histograma de los residuos muestra claramente que no se distribuyen de manera normal.



### Tests de Portmanteau de autocorrelación

Para analizar de manera más formal la presencia o ausencia de autocorrelación en los residuos, podemos realizar estas pruebas estadísticas para determinar si las primeras $h$ autocorrelaciones son significativamente distintas de cero o no.

#### Test de Box-Pierce

$$
Q=T \sum_{k=1}^{h} r_{k}^{2}
$$
En este test, $h$ es el rezago máximo a considerar y $T$ es la cantidad de observaciones en la muestra.

Si cada $r_{k}^{2}$ es pequeña, entonces $Q$ será pequeña. Se sugiere utilizar $h = 10$ para datos no estacionales y $h = 2m$ para datos estacionales (donde $m$ es el periodo estacional). Sin embargo, la prueba no es tan buena cuando $h$ es grande, relativamente (o sea, cuando $h$ es mayor a $T/5$). En esos casos, es mejor utilizar $h = T/5$.

#### Test de Ljung-Box

Un test relacionado y que, generalmente, es más preciso es el test de Ljung-Box.

$$
Q^{*}=T(T+2) \sum_{k=1}^{h}(T-k)^{-1} r_{k}^{2}
$$
En este caso es igual: valores grandes de $Q^{*}$ son indicios de que las autocorrelaciones no provienen de ruido blanco.

> Entonces, la hipótesis nula de estas pruebas es que la serie en cuestión **no está autocorrelacionada**. En otras palabras, la $H_0$ dice que **la serie es ruido blanco**. Si $\alpha$ es el nivel de significancia (el nivel máximo de error que estamos dispuestos a aceptar) y si el ¨*p-value* $< \alpha$, entonces **rechazamos $H_0$**, de lo contrario, no rechazamos la $H_0$.

Las pruebas para el modelo ingenuo:

```{r paged.print=FALSE}
# lag=h and fitdf=K
aug %>% features(.resid, box_pierce, lag=10, dof=0)

aug %>% features(.resid, ljung_box, lag=10, dof=0)

```

Estas pruebas nos indican que los residuos del modelo Naïve sí son ruido blanco (no están autocorrelacionados).

Para el modelo de la media:

```{r paged.print=FALSE}
google_2015 %>% 
  model(MEAN(Close)) %>% augment() %>% 
  features(.resid, box_pierce, lag = 10, dof = 0)

google_2015 %>% 
  model(MEAN(Close)) %>% augment() %>% 
  features(.resid, ljung_box, lag = 10, dof = 0)
```


En ambas pruebas, el p-value resulta ser muy alto, por lo que no podemos distinguir los residuales del ruido blanco.


# Intervalos de predicción

Un intervalo de predicción se puede escribir como

$$\hat{y}_{T+h | T} \pm c \hat{\sigma}_{h}$$
donde $c$ es el porcentaje de cobertura de probabilidad. Normalmente utilizaremos 80% y 95%, pero se puede utilizar cualquier porcentaje.

La utilidad de los intervalos de predicción yace en el hecho de que nuestros pronósticos tienen cierta incertidumbre. Así, entre mayor sea el intervalo, menos preciso será nuestro pronóstico, y vice versa.

Un pronóstico puntual por si solo no sirve de gran cosa. Es necesario acompañarlo de su intervalo de predicción.

### Intervalos de predicción de un paso

Cuando se realizan pronósticos de un paso (*one-step forecast*), la desviación estándar del pronóstico es prácticamente la misma que la desviación estándar de los residuos.

### Intervalos de predicción de paso múltiple (*multi-step*)

Conforme se va aumentando el horizonte de pronostico, el intervalo de predicción tiende a aumentar. Entre más adelante en el tiempo queramos pronosticar, tendremos mayor incertidumbre (no es lo mismo querer predecir el tipo de cambio para mañana, que el de diciembre, p. ej.). Esto es, $\sigma_h$ incrementa con $h$. Entonces, requerimos estimaciones de $\sigma_h$.

Para el caso del one-step forecast, ya vimos que podemos tomar la desviación estándar de los residuos como estimación de $\sigma_h$. Para el caso multi-step, asumimos que los residuos no están autocorrelacionados y se requieren métodos de cálculo un poco más complejos.

### Métodos de referencia

Si $\hat{\sigma}$ es la desviación estándar de los residuos y $\hat{\sigma}_{h}$ es la desviación estándar del pronóstico $h$-step, podemos calcular para cada método de pronóstico de referencia:

**Pronósticos de media:** $\hat{\sigma}_{h}=\hat{\sigma} \sqrt{1+1 / T}$

**Pronósticos naïve:** $\hat{\sigma}_{h}=\hat{\sigma} \sqrt{h}$

**Pronósticos naïve estacionales:** $\hat{\sigma}_{h}=\hat{\sigma} \sqrt{k+1}$, donde $k$ es la parte entera de $(h-1)/m$.

**Pronósticos de drift:** $\hat{\sigma}_{h}=\hat{\sigma} \sqrt{h(1+h / T)}$.

Utilizando la paquetería `fable`, es muy sencillo obtener pronósticos y sus bandas.

```{r eval=FALSE, paged.print=FALSE}
google_2015 %>%
  model(NAIVE(Close)) %>%
  forecast(h = 10) %>%
  hilo()
```

Esto mismo se puede graficar, como lo hemos hecho anteriormente:

```{r}
google_2015 %>%
  model(NAIVE(Close)) %>%
  forecast(h = 10) %>%
  autoplot(google_2015)
```

### Intervalos de predicción con residuales bootstrap

Cuando no es razonable asumir **normalidad** en los residuos, podemos aplicarles *bootstraping*, ya que esto solo asume la no autocorrelación.

Teníamos que los residuos se calculan $e_{t}=y_{t}-\hat{y}_{t | t-1}$. Reescribiendo, tenemos que:

$$y_{t}=\hat{y}_{t | t-1}+e_{t}$$
y podemos simular la siguiente observación de una serie de tiempo con:

$$y_{T+1}=\hat{y}_{T+1 | T}+e_{T+1}$$
 
donde $\hat{y}_{T+1} | T$ es el pronóstico de un periodo (one-step) y $e_{T+1}$ es el error futuro (que desconocemos). Ya que no están autocorelacionados los errores, y puesto que asumimos que los errores futuros serán similares a los históricos, podemos cambiar $e_{T+1}$ al hacer un muestreo de los residuos. Podemos realizar el mismo proceso para $y_{T+2}, y_{T+3}, \dots$.

Si realizamos esto varias veces, obtendremos muchos escenarios futuros posibles. Para ver algunos de ellos, utilizamos `generate`.

![Dr. Strange simulando mediante bootstrap 14,000,605 escenarios posibles](../images/dr_strange.jpeg)

```{r bootstrap}
fit <- google_2015 %>%
  model(NAIVE(Close))

sim <- fit %>%  generate(h = 30, times = 5, bootstrap = TRUE, seed = 123) # only works with dev version

sim
```




Lo que hicimos fue generar 5 escenarios futuros posibles (`times = 5`) para los siguientes 30 días de trading (`h = 30`). Si graficamos esto, tenemos:

```{r bootstrap plot}
google_2015 %>%
  ggplot(aes(x = day)) +
  geom_line(aes(y = Close), size = 1) +
  geom_line(aes(y = .sim, colour = as.factor(.rep)), data = sim, size = 1) +
  ggtitle("Google closing stock price") +
  guides(col = FALSE)
```

Con esto, podemos obtener intervalos de predicción, al calcular los percentiles de los escenarios futuros. El resultado se llama **intervalo de predicción bootstrapped**. Esto se puede lograr fácilmente con `forecast`.

```{r}
fc <- fit %>% forecast(h = 30, bootstrap = TRUE)
fc
```

Graficando:


```{r}
fc %>% autoplot(google_2015) +
  ggtitle("Google closing stock price")
```
## Tarea predicción de producción de ladrillos

```{r}
aus_production %>% 
  autoplot(Bricks)

aus_production %>% 
  autoplot(box_cox(Bricks, lambda = 0.4))

aus_production %>% 
  select(Quarter, Bricks) %>% 
  drop_na() %>% 
  slice_tail(n = 8)

# Los datos de entrenamiento
bricks_train <- aus_production %>% 
  filter_index(. ~ "2003 Q2")# %>% 
  # mutate(log_bricks = log(Bricks)) # no deberían hacer la transformación antes del modelo

# Estimación de modelos
bricks_fit <- bricks_train %>% 
  model(
    Media                = MEAN(log(Bricks)),
    Ingenuo              = NAIVE(log(Bricks)),
    `Ingenuo estacional` = SNAIVE(log(Bricks)),
    Deriva               = RW(log(Bricks) ~ drift())
  )

bricks_aug <- augment(bricks_fit)
bricks_aug

graf_residuos <- function(modelo = "Media"){
  bricks_fit %>% 
    select(modelo) %>% 
    gg_tsresiduals() +
    ggtitle(paste("Diagnóstico de residuales para el modelo", modelo))
}

graf_residuos("Media")
graf_residuos("Ingenuo")
graf_residuos("Ingenuo estacional")
graf_residuos("Deriva")

bricks_fc <- bricks_fit %>% 
  forecast(h = "2 years")

bricks_fc %>% 
  autoplot(aus_production %>% select(Quarter, Bricks) %>% drop_na(), level = NULL)

bricks_fc %>%
  autoplot(aus_production %>% select(Quarter, Bricks) %>%  drop_na() %>% filter_index("1995 Q1" ~ .),
           level = NULL)

bricks_fc %>%
  filter(.model == "Ingenuo estacional") %>%
  autoplot(aus_production %>% select(Quarter, Bricks) %>%  drop_na() %>% filter_index("1995 Q1" ~ .))
```



# Tarea

1. Conseguir datos históricos sobre dos series de tiempo.
2. Seguir los pasos del flujo de trabajo de pronóstico.
3. Estimar los modelos de referencia (*benchmark*) que consideren adecuados para su serie.
4. Realicen el diagnóstico de residuales e interpreten los resultados.
5. Ejecuten un pronóstico (ustedes deciden el horizonte de pronóstico) y definan si utilizar el método bootstrap o no y justifiquen su decisión.

***

# Pronósticos con transformaciones

¿Qué sucede cuando realizamos pronósticos de series a las que les hicimos alguna transformación? Por ejemplo, si los contratan para pronosticar las ventas de cubrebocas en la farmacia y ustedes hubieran realizado una transformación logarítmica de los datos, ¿cómo se realiza el pronóstico?

Cuando se realiza una transformación matemática, el pronóstico se hace con esa serie transformada, y posteriormente tenemos que darle *reversa* a la transformación (*back-transformation*), para obtener pronósticos en la escala original de la serie.

La transformación inversa de Box-Cox está dada por:

$$
y_{t}=\left\{\begin{array}{ll}
\exp \left(w_{t}\right) & \text { si } \lambda=0 \\
\left(\lambda w_{t}+1\right)^{1 / \lambda} & \text { en otro caso }
\end{array}\right.
$$

La paquetería `fable` convenientemente realiza la transformación inversa en automático, cuando se especifica en la definición del modelo.

### Intervalos de predicción con transformaciones

El intervalo de predicción de una serie transformada se calcula, primeramente, en la escala transformada, y posteriormente se hace la transformación inversa a la escala original. Hacer esto mantiene los porcentajes de cobertura de probabilidad originales, pero el resultado ya no es simétrico alrededor de la estimación puntual.

Es importante mencionar que las transformaciones tienen muy poco efecto en la estimación puntual, pero pueden llegar a tener un gran impacto en los intervalos de predicción.

### Ajustes por sesgo

Un problema al realizar transformaciones matemáticas, como Box-Cox es que la estimaciones puntuales re transformadas ya no representan la **media** de la distribución de predicción, sino que representan ahora la **mediana**. En muchos casos puede no ser tan grave esto, pero en ocasiones el pronóstico promedio es requerido:

* *P. ej., si quieren realizar el pronóstico de la venta de cubrebocas en las farmacias de la ZMG, para, al sumarlos, obtener el pronóstico de las ventas totales de cubrebocas en ZMG. La suma de las medias da como resultado el total, pero la suma de medianas no.*

La transformación inversa de la media, para Box-Cox es:

$$
y_{t}=\left\{\begin{array}{ll}
\exp \left(w_{t}\right)\left[1+\frac{\sigma_{h}^{2}}{2}\right] & \text { si } \lambda=0 \\
\left(\lambda w_{t}+1\right)^{1 / \lambda}\left[1+\frac{\sigma_{h}^{2}(1-\lambda)}{2\left(\lambda w_{t}+1\right)^{2}}\right] & \text { en otro caso }
\end{array}\right.
$$
donde $\sigma_{h}^{2}$ es la varianza del pronóstico en el horizonte-$h$ en la escala transformada. Entre más grande sea la varianza, mayor será la diferencia entre la media y la mediana. A esto se le conoce como la estimación **ajustada por sesgo**.

Veamos un ejemplo sobre el pronóstico del precio promedio del huevo.

```{r}
eggs <- as_tsibble(fma::eggs)
eggs %>% 
  model(RW(log(value) ~ drift())) %>% 
  forecast(h=50) %>% 
  autoplot(eggs, level = 80,  
           point_forecast = lst(mean, median))
```

La línea continua representa el pronóstico de la media (*ajustado por sesgo*), mientras que la línea punteada muestra el pronóstico de la mediana (*sin corrección por sesgo). `fable` en automático nos produce pronósticos ajustados por sesgo. Como se ve en el código, para generar pronósticos sesgados, tenemos que especificarlo mediante `point_forecast = lst(median)`.

# Pronósticos con descomposición

La descomposición de series de tiempo puede ser útil para producir pronósticos. Reescribiendo las fórmulas de descomposición aditiva y multiplicativa:

$$y_{t}=\hat{S}_{t}+\hat{A}_{t}$$

donde $\hat{A}_{t}$ es la serie desestacionalizada ( $\hat{A}_{t} = \hat{T}_{t} + \hat{R}_{t}$).

$$y_{t}=\hat{S}_{t}\hat{A}_{t}$$

con  $\hat{A}_{t} = \hat{T}_{t}  \hat{R}_{t}$.


Así, el pronóstico se realiza en dos pasos: un pronóstico para el componente estacional, y un pronóstico separado para la serie desestacionalizada. De hecho, el pronóstico del componente estacional es simplemente el método **naïve estacional**. Para los datos desestacionalizados, podemos utilizar cualquier modelo de pronóstico que veremos más adelante.


```{r}
us_retail_employment <- us_employment %>%
  filter(year(Month) >= 1990, Title == "Retail Trade")

us_retail_employment %>% 
autoplot(Employed)

dcmp <- us_retail_employment %>%
  model(STL(Employed ~ trend(window = 7), robust=TRUE)) %>%
  components() %>%
  select(-.model)

dcmp

dcmp %>%
  model(NAIVE(season_adjust)) %>%
  forecast() %>%
  autoplot(dcmp) + ylab("New orders index") +
  ggtitle("Pronóstico naïve de los datos desestacionalizados")
```

A esta serie, podemos agregarle nuevamente la estacionalidad con la función `decomposition_model()`.


```{r}
us_retail_employment %>%
  model(stlf = decomposition_model(
             STL(Employed ~ trend(window = 7), robust = TRUE),
             NAIVE(season_adjust),
             SNAIVE(season_year)
  )) %>%
  forecast() %>%
  autoplot(us_retail_employment)
```

# Evaluación del desempeño de los pronósticos

### Conjuntos de entrenamiento y prueba

Como hemos dicho, es muy importante separar nuestros datos en dos conjuntos: un conjunto de datos de **entrenamiento**, que son los que se utilizan para estimar el modelo, y un conjunto de datos de **prueba**, donde se evalúa el desempeño del pronóstico.

El tamaño de la prueba es, generalmente, del 20% del total de datos disponibles, aunque también puede depender del horizonte de pronóstico requerido. La prueba tiene que ser al menos tan grande como el horizonte de pronóstico más largo que se requiera (si se necesitan pronósticos para todo el siguiente año, la prueba tiene que ser del tamaño de al menos de todo el siguiente año).

**NOTA**: Es importante tener en cuenta lo siguiente:

* Un modelo que se ajusta muy bien a los datos de entrenamiento no necesariamente produce los mejores pronósticos.

* Podemos llegar a tener un ajuste perfecto del modelo a los datos, si aumentamos la cantidad de parámetros.

* Puede darse un efecto de sobre ajuste (mejor conocido como *over-fitting*) y esto es tan malo como tener un muy mal ajuste.

En alguna literatura o paqueterías pueden encontrar que al conjunto de datos de entrenamiento se les conozca como "*in-sample data*" y al conjunto de prueba "*out-of-sample data*".

### Funciones para segmentar las series de tiempo

Hemos visto que podemos utilizar la función `filter` para filtrar una base de datos o una serie de tiempo. 

```{r}
aus_production
```
Por ejemplo, tomando la producción en Australia, podemos filtrar para tener los datos a partir de 1995:

```{r}
aus_production %>% filter(year(Quarter) >= 1995)
```

o para obtener los datos de una cierta estación:

```{r}
aus_production %>% filter(quarter(Quarter) == 1)
```

Otra función útil para filtrar o segmentar series de tiempo es `slice()`, que utiliza el índice para filtrar los datos.

```{r}
aus_production %>%
  slice(n()-19:0)
```
Esto filtra los últimos 20 datos (5 años).

A partir de `dplyr 1.0.0`, existen algunas variantes de `slice()` que pueden sernos bastante útiles, como `slice_head()`, `slice_tail()`, `slice_sample()`, `slice_max()`, etc.

Podemos reescribir el código de arriba para hacerlo más explícito:

```{r}
aus_production %>% 
  slice_tail(n = 20)
```

Podemos usar `slice` para datos agrupados:

```{r}
aus_retail %>%
  group_by(State, Industry) %>%
  slice(1:12)
```
Otra función es `top_n`, la cual nos permite obtener las `n` observaciones más extremas.

```{r}
gafa_stock %>%
  group_by(Symbol) %>%
  top_n(1, Close)
```

## Errores de pronóstico

El error de pronóstico es la diferencia entre el valor real ocurrido y el dato pronosticado. 

$$e_{T+h}=y_{T+h}-\hat{y}_{T+h | T}$$

Los errores de pronóstico son distintos de los residuales en dos formas:

1. Los residuales se calculan con los datos de *entrenamiento*, mientras que los errores de pronóstico se calculan con los de *prueba*.
2. Los residuos se calculan mediante pronósticos de un paso (one-step), donde los errores de pronóstico pueden ser multi-step.

Hay muchos tipos de cálculo del error general de un modelo de pronóstico.

### Errores dependientes de la escala de los datos

Los errores de pronóstico están medidos en la misma escala de los datos originales. Hay ciertos tipos de error que solo se basan en
$e_{T}$, por lo que también dependen de la escala y no pueden ser utilizados para comparar el desempeño con otra serie de tiempo que tenga otras unidades.

Los dos más utilizados en este rubro son el **MAE** y el **RMSE**.

$$
\begin{aligned}
&\text { Mean absolute error: } \mathrm{MAE}=\operatorname{mean}\left(\left|e_{t}\right|\right)\\
&\text { Root mean squared error: } \operatorname{RMSE}=\sqrt{\operatorname{mean}\left(e_{t}^{2}\right)}
\end{aligned}
$$

El MAE es muy utilizado debido a su facilidad de cómputo y de interpretación. Un método de pronóstico que minimiza el MAE nos dará pronósticos de la mediana de la distribución. Pronósticos que minimizan el RMSE obtienen pronósticos de la media, por lo que este método también es muy utilizado, a pesar de ser más pesado computacionalmente y complicado de interpretar.

### Errores porcentuales

Los errores porcentuales, al ser un porcentaje, no tienen unidades y son utilizados para comparar el desempeño de pronósticos de distintos conjuntos de datos. El más utilizado es el MAPE (error absoluto promedio porcentual). Si definimos al error porcentual como $p_{t}=100 e_{t} / y_{t}$

Mean absolute percentage error: **MAPE** $=\operatorname{mean}\left(\left|p_{t}\right|\right)$

La desventaja con estos errores es que se indeterminana o vuelven infinitos con valores de $y_t = 0$. Para ello, se definió el MAPE simétrico:

$$\operatorname{sMAPE}=\operatorname{mean}\left(200\left|y_{t}-\hat{y}_{t}\right| /\left(y_{t}+\hat{y}_{t}\right)\right)$$

Aunque este método no se recomienda tanto utilizarlo en la práctica.

### Errores escalados

Una alternativa a los errores porcentuales cuando se quieren comparar los pronósticos de distintas series de tiempo so los **errores escalados**, propuestos por Hyndman & Koehler. En éstos, se escalan los errores con base en el MAE de entrenamiento de utilizar un modelo Naïve o Seasonal Naïve, dependiendo de si los datos tienen estacionalidad.

Para series no estacionales, los errores escalados serían:

$$
q_{j}=\frac{e_{j}}{\frac{1}{T-1} \sum_{t=2}^{T}\left|y_{t}-y_{t-1}\right|},
$$
mientras que para series estacionales, los errores escalados estarían dados por:

$$
q_{j}=\frac{e_{j}}{\frac{1}{T-m} \sum_{t=m+1}^{T}\left|y_{t}-y_{t-m}\right|}.
$$

De esta forma podemos obtener el **Error Medio Absoluto Escalado (MASE)** con

$$
\operatorname{MASE} = \operatorname{mean}(|q_j|).
$$


---

```{r}
recent_production <- aus_production %>% filter(year(Quarter) >= 1992)
beer_train <- recent_production %>% filter(year(Quarter) <= 2007)

beer_fit <- beer_train %>%
  model(
    Mean = MEAN(Beer),
    `Naïve` = NAIVE(Beer),
    `Seasonal naïve` = SNAIVE(Beer),
    Drift = RW(Beer ~ drift())
  )

beer_fc <- beer_fit %>%
  forecast(h = 10)

beer_fc %>%
  autoplot(recent_production, level = NULL) +
  xlab("Year") + ylab("Megalitres") +
  ggtitle("Forecasts for quarterly beer production") +
  guides(colour=guide_legend(title="Forecast"))
```
La función `accuracy()` nos permite calcular varias de estas métricas de error de manera muy sencilla. Incluso, podemos calcularlas para el ajuste del modelo a los datos de entrenamiento, o calcular el error de pronóstico. Todo depende de los datos que le pasemos a la función.

Si utilizamos la tabla de modelos (`mable`), obtendremos las métricas de error para el ajuste del modelo a sus datos de entrenamiento:

```{r model accuracy}
accuracy(beer_fit)
```
Las métricas de error para el ajuste a los datos de entrenamiento nos pueden servir para decidir qué modelos utilizar para pronosticar.


En cambio, si utilizamos la tabla de pronósticos (`fable`), junto con los datos de validación (o datos de prueba), obtendremos los errores de pronóstico:

```{r forecast accuracy}
beer_fc %>% 
  accuracy(recent_production)
```

Las métricas de error sobre los datos de prueba nos dan una mayor claridad de cuáles modelos parecen producir los mejores pronósticos.

Una vez decidido esto, se recalculan los modelos utilizando **toda** la información histórica (es decir, sin separar en conjuntos de entrenamiento y prueba), y se producen los pronósticos reales a presentar.
